Spectral Spaces in o-minimal and other NIP theories

TL;DR

This paper explores spectral topologies in NIP theories, linking o-minimal spectra with other topological spaces, and introduces the honest topology to analyze invariant types, revealing new structural insights.

Contribution

It establishes connections between spectral topologies and types in NIP theories, introduces the honest topology, and characterizes the space of invariant types in o-minimal and general NIP settings.

Findings

The space of closed points in definably compact groups is homeomorphic to the space of infinitesimal types.

Invariant types form a normal spectral space with finitely satisfiable types as closed points.

The honest topology makes the set of invariant types a normal spectral space with properties aligning with known retractions.

Abstract

We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction.